Topology: Sphere to plane
I guess this is one way to do it:

Delete the top center vertex and extrude out the sides.  Remove Doubles:







Select the horizontal edges only on the faces you just created and Subdivide them twice:







Use Proportional Editing to bring the subdivided loops toward the corner a little more:







Now select the only the edges shown and subdivide those as well.  This will make the odd number of loops that we need to complete the Quad structure on the top and bottom.







Next, create the edges shown below and subdivide the edges once.  Press GG to slide the new vertices toward each other and merge them at the center.  Subdivide the remaining edges twice more (Note: I have a the number keys 1-3 mapped to Vertex, Edge, and Face selection).  You can use the F2 Add-on to make face creation faster:







Continue that process until you get to the end and simply extrude out the remaining vertices to create the final quads:







Repeat the same process above for the top as well.  Since we cheated at the beginning by deleting the center vertex, you can use Proportional Editing to add a little smoother termination.  Hide the geometry below so that it's unaffected:







Here's the final.  It may not be the best way, but it's one way to do it:







EDIT: A quick and dirty way to do it, building on @Marcus' answer, is to take the subdivided cube and make "projections" for X, Y, and Z.  You can then dissolve unnecessary edge loops.  It doesn't really teach much about modeling, but it is does make an eighth of a sphere out of quads: